Asymptotic Average SER - Performance Analysis

4.3 Performance Analysis

4.3.2 Asymptotic Average SER

4. WP DF Relay System under Nakagami-m Fading: EH at Relay Node

where P_e(γ_ϕ₂) is the conditional SER of ϕ₂ ∈ {SR, RD} link and γ_ϕ₂ is the corresponding instantaneous SNR defined as in (4.8a) and (4.8b). The end-to-end average SER is obtained by taking expectation of (4.28) as

P_e = E[P_e(γ_SR, γ_RD)]

= 1−(1−P_e,SR)(1−P_e,RD), (4.29)

where P_e,SR=E[P_e(γ_SR)] and P_e,RD =E[P_e(γ_RD)]. P_e,SR is given in (4.23) for ϕ₁ =SR and P_e,RD is obtained using the MGF based approach. The PDF of the instantaneous link SNR γ_RD = 2(t₂)², where t₂ =ν₂|h_SR||h_RD|is obtained using (4.16) as

f_γRD(γ_RD) = m_SRm_RD

Γ(m_SR)Γ(m_RD)¯γ_RDG^2,0_0,2

m_SRm_RDγ_RD

¯ γ_RD

−

m_SR−1, m_RD−1

, (4.30)

where ¯γ_RD = 2¯γ₂. The corresponding MGF is obtained on substituting (4.30) in (4.21) and following the steps adopted in Appendix A.1 as

M_γRD(s) = m_SRm_RD

Γ(mSR)Γ(mRD)s¯γ_RDG^2,0_0,2

m_SRm_RD s¯γRD

mSR−1, mRD−1

. (4.31)

An approximate and computationally efficient expression for the average SER of RD link can be analyzed on substituting (4.31) in (4.26) forϕ=RD. Therefore,P_e,RD is

P_e,RD ≈ a_Mm_SRm_RD πΓ(m_SR)Γ(m_RD)

Z _π/2

sin²(θ) g¯γ_RD G^2,0_0,2

m_SRm_RDsin²(θ) g¯γ_RD

m_SR−1, m_RD−1

. (4.32)

A closed-form solution of (4.32) can be obtained using (4.13) and (A.11). Thus, we have

P_e,RD ≈ a_Mm_SRm_RD

2√πΓ(m_SR)Γ(m_RD)g¯γ_RDG^2,2_2,3

m_SRm_RD g¯γ_RD

0,−1/2

m_SR−1, m_RD−1,−1

. (4.33)

An approximate expression of the end-to-end average SER is obtained on substituting (4.23) and (4.33) in (4.29).

4.3 Performance Analysis

4.3.2.1 With SD Link

The end-to-end average SER in (4.19) for ¯γ_ϕ → ∞ is given by

P_e^∞ ≃ P_e,SR^∞ P_e,SD^∞ +P_e,EGC^∞ , (4.34)

whereP_e,SR^∞ ,P_e,SD^∞ , andP_e,EGC^∞ are the high SNR approximations ofP_e,SR,P_e,SD, andP_e,EGC, respectively. Now, P_e,SD^∞ and P_e,SR^∞ can be obtained using the high SNR approximation of MGF in (4.22), that is, Mϕ1(s)≈(s¯γϕ1/mϕ1)^−m^ϕ¹ and (4.26) as

P_e,ϕ^∞₁ ≈ aMΓ(1/2 +mϕ1) 2√

πΓ(1 +m_ϕ₁)

mϕ1

g¯γ_ϕ₁ mϕ1

. (4.35)

In order to determine P_e,EGC^∞ , we approximate PDF and MGF of γ_EGC at high SNR. Since EGC combines signal received via independentSDandRDlinks, PDF and MGF ofγ_EGCare approximated at high SNR for each link individually.

The PDF in (4.16) is rewritten using (B.4) as

f_t₂(t₂) = 4 (t₂)⁽^mSR⁺^mRD⁻¹⁾ Γ(mSR)Γ(mRD)

m_SRm_RD

¯ γ2

mSR⁺mRD

K_mSR₋_mRD

s4m_SRm_RD(t₂)²

¯ γ2

! ,(4.36)

where K_r(y) is ther-th order modified Bessel’s function of the second kind. Now, using the relation (B.6), (4.36) can be approximated for ¯γ₂→ ∞. We find two cases while doing the approximation: a) (m_SR−m_RD) = 0 and b) |m_SR−m_RD|>0.

a) For (m_SR−m_RD) = 0: Let m_SR = m_RD = m_a, the approximation of (4.36) realized using (B.6) is

f_t^∞₂ (t2) ≈ −4 (t2)^(2m^a⁻¹⁾ (Γ(m_a))²

(ma)²

¯ γ₂



 s

4ma2(t2)²

¯ γ₂



. (4.37)

Using (4.37) and A.6, an approximation of f_γ_EGC(γ_EGC) is analyzed in Appendix A.4 as f_γ^∞

EGC(γ_EGC) ≈ 2Γ(1 + 2m_SD)Γ(2m_a)(m_SD+m_a) Γ(1 +mSD)Γ(1 + 2mSD+ 2ma)(Γ(ma))²

m_SD

¯ γ1

_mSD

(m_a)²

¯ γ2

!ma

× 2Dn− 1

(mSD+ma)−ln 4G²(m_a)²γ_EGC

¯ γ2

(γ_EGC)⁽^mSD^+m^a⁻¹⁾, (4.38)

where ln(G²) = P^∞

n=1

(2(2m_a−1))/(n(n+ 2m_a−1)) andDn= (2m_SD+2m_a) P^∞

n=1

(Γ(n+ 2m_SD+ 2m_a))

4. WP DF Relay System under Nakagami-m Fading: EH at Relay Node

/(nΓ(1 +n+ 2m_SD+ 2m_a)). The corresponding MGF is determined using (4.24), (4.38), (B.7) and (B.9) as

M_γ^∞

EGC(s) ≈ 2Γ(1 + 2m_SD)Γ(2m_a)Γ(1 +m_SD+m_a) Γ(1 +m_SD)Γ(1 + 2m_SD+ 2m_a)(Γ(m_a))²

m_SD s¯γ₁

_mSD

(m_a)² s¯γ₂

!ma

× 2Dn−ψ(m_SD+m_a)− 1

(m_SD+m_a) −ln 4G²(m_a)² s¯γ₂

, (4.39)

whereψ(·) is digamma function [170, eq. (6.3.1)]. The high SNR approximation ofPe,EGCis obtained using (4.26) and (4.39) followed by some algebraic manipulations. The resultant expression is

P_e,EGC^∞ ≈ a_MΓ(1 + 2m_SD)Γ(2m_a)Γ(1/2 +m_SD+m_a)

√πΓ(1 +m_SD)Γ(1 + 2m_SD+ 2m_a)(Γ(m_a))²

× m_SD

g¯γ₁ _mSD

(m_a)² g¯γ₂

!ma

−ψ(m_SD+m_a+ 1/2) +ψ(m_SD+m_a+ 1)

−ψ(m_SD+m_a)− 1

(m_SD+m_a) −ln 4G²(m_a)² g¯γ₂

! + 2Dn

. (4.40)

We have used the relations Rπ/2

0 (sin²(θ))^m^SD⁺^m^adθ= (√πΓ(m_SD+m_a+ 1/2))/(2Γ(m_SD+m_a+ 1)) and (B.10) to obtain the outcome presented in (4.40).

b) For|m_SR−m_RD|>0: In this case, an approximation of (4.36) is obtained using (B.6) as

f_t^∞₂ (t₂) ≈ 2Γ(|mSR−mRD|) Γ(m_SR)Γ(m_RD)

mSRmRD

¯ γ₂

(t₂)^2m^b⁻¹, (4.41)

wherem_b = (m_SR+m_RD− |m_SR−m_RD|)/2. Next, we obtainf_γ_EGC(γ_EGC) using (4.41) and adopting steps followed for analyzing (4.38) as

f_γ^∞

EGC(γ_EGC) ≈ Γ(1 + 2mSD)Γ(|mSR−mRD|)Γ(2m_b) Γ(1 +m_SD)Γ(m_SR)Γ(m_RD)Γ(2m_SD+ 2m_b)

× m_SD

¯ γ₁

_mSD

m_SRm_RD

¯ γ₂

(γ_EGC)^mSD^+m^b⁻¹. (4.42)

Using (4.24) and (4.42), the approximate expression of MGF is written as

M_γ^∞

EGC(s) ≈ Γ(1 + 2m_SD)Γ(|m_SR−m_RD|)Γ(2m_b) Γ(1 +m_SD)Γ(m_SR)Γ(m_RD)Γ(2m_SD+ 2m_b)

×Γ(m_SD+m_b)

m_SD s¯γ1

_mSD

m_SRm_RD s¯γ2

_mRD

. (4.43)

4.3 Performance Analysis

P_e,EGC^∞ can be analyzed using (4.26) and (4.43) as

P_e,EGC^∞ ≈ a_MΓ(1 + 2m_SD)Γ(|m_SR−m_RD|)Γ(2m_b)Γ(m_SD+m_b) 2πΓ(1 +m_SD)Γ(m_SR)Γ(m_RD)Γ(2m_SD+ 2m_b)

×B 1

2, m_SD+m_b+1 2

m_SD g¯γ₁

_mSD

m_SRm_RD g¯γ₂

, (4.44)

where B(x, y) = Γ(x)Γ(y)/Γ(x+y) is beta function.

The high SNR approximations ofP_e,EGC^∞ for (m_SR−m_RD) = 0 and|m_SR−m_RD|>0 in (4.40) and (4.44), respectively are unified as

P_e,EGC^∞ ≈ A¹(B¹+ρ1ln(¯γ2))

(¯γ₁)^mSD(¯γ₂)^µ¹ . (4.45) A1,B1,µ₁, and ρ₁ are defined for each case as follows

a) For (m_SR−m_RD) = 0: µ₁ =m_a,ρ₁ = 1,

A¹ = a_MΓ(1 + 2m_SD)Γ(2m_a)Γ(1/2 +m_SD+m_a)(m_SD)^mSD(m_a)^2m^a

√πΓ(1 +m_SD)Γ(1 + 2m_SD+ 2m_a)(Γ(m_a))²g^mSD^+m^a

and

B1 =

−ψ(m_SD+m_a+ 1/2) +ψ(m_SD+m_a+ 1)− ψ(m_SD+m_a)

− 1

(m_SD+m_a)−ln

4G²(ma)² g

+ 2Dⁿ

b) For|m_SR−m_RD|>0: µ₁ =m_b,ρ₁= 0, B1= 1 and

A¹ = B(1/2, mSD+m_b+ 1/2) (mSD)^mSD(mSRmRD)^mSR

× a_MΓ(1 + 2m_SD)Γ(|m_SR−m_RD|)Γ(2m_b)Γ(m_SD+m_b) 2πΓ(1 +m_SD)Γ(m_SR)Γ(m_RD)Γ(2m_SD+ 2m_b)g^mSD^+m^b.

Using (4.34), (4.35) and (4.45), asymptotic expression of the end-to-end average SER is given by

P_e^∞ ≈ Z1

(¯γ_SD)^mSD(¯γ_SR)^mSR +A1(B1+ρ₁ln(¯γ₂))

(¯γ₁)^mSD(¯γ₂)^µ¹ , (4.46) where

Z1 = (a_M)²Γ(1/2 +m_SD)Γ(1/2 +m_SR)(m_SD)^mSD(m_SR)^mSR 4πΓ(1 +m_SD)Γ(1 +m_SR)g^mSD⁺^mSR .

4. WP DF Relay System under Nakagami-m Fading: EH at Relay Node

4.3.2.2 Without SD Link

The high SNR approximation of the end-to-end SER in (4.29) is given by

P_e^∞ ≃ P_e,SR^∞ +P_e,RD^∞ , (4.47)

where P_e,SR^∞ and P_e,RD^∞ are the high SNR approximation of P_e,SR and P_e,RD, respectively. P_e,SR^∞ is same as (4.35) for ϕ₁ =SR and P_e,RD^∞ is determined using the high SNR approximation of the PDF expression in (4.30). Now, to determine P_e,RD^∞ ,f_γRD^∞ (γ_RD) is obtained using (4.30), (B.4) and (B.6) for the cases: a) (m_SR−m_RD) = 0 and b) |m_SR−m_RD|>0 as

f_γRD^∞ (γ_RD) ≈ −(γ_RD)^m^a⁻¹ (Γ(m_a))²

(m_a)²

¯ γ_RD

!ma

ln 4(m_a)²γ_RD

¯ γ_RD

(4.48)

and

f_γ^∞

RD(γ_RD) ≈ Γ(|mSR−mRD|)(γRD)^m^b⁻¹ Γ(mSR)Γ(mRD)

mSRmRD

¯ γRD

, (4.49)

respectively. The corresponding average SER expressions for cases a) and b) can be analyzed on adopting the steps followed to analyze (4.40) and (4.44), respectively. Thus, the unified expression of P_e,RD^∞ for the two cases can be written as

P_e,RD^∞ ≈ A2(B2+ρ₂ln(¯γ_RD))

(¯γ_RD)^µ² . (4.50)

A2,B2,ρ₂, and µ₂ are defined for each case as follows

a) For (m_SR−m_RD) = 0 : µ₂ = m_a, ρ₂ = 1, A2 = (a_MΓ(1/2 +m_a)(m_a)^2m^a)/(2√

πΓ(m_a)Γ(1 + ma)g^m^a), andB²=−ψ(ma)−ψ(ma+ 1/2) +ψ(ma+ 1)−ln(4(ma)²/g).

b) For|mSR−mRD|>0: µ2 =m_b,ρ2 = 0,B² = 1, and

A²= aMΓ(|mSR−mRD|)Γ(1/2 +m_b)Γ(m_b)(mSRmRD)^m^b 2√

πΓ(m_SR)Γ(m_RD)Γ(1 +m_b)g^m^b .

The asymptotic expression for the end-to-end average SER is obtained using (4.35), (4.47) and (4.50) as

P_e^∞ ≈ Z2

(¯γ_SR)^mSR + A2(B2+ρ₂ln(¯γ_RD))

(¯γ_RD)^µ² , (4.51)

where

Z2 = a_MΓ(1/2 +m_SR)(m_SR)^mSR 2√

πΓ(1 +mSR)g^mSR .

Dalam dokumen SPEECH ENHANCEMENT (Halaman 96-101)